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Existence of positive solutions to \(p\)-Kirchhoff-type problem without compactness conditions. (English) Zbl 1317.35058
Summary: In this paper, we study the existence of positive solutions to the \(p\)-Kirchhoff-type problem \[ \begin{cases} \Bigg(a+\lambda\Bigg(\int_{\mathbb R^N}(|\nabla u|^p+b|u|^p)dx\Bigg)^\tau\Bigg)(-\Delta_pu+b|u|^{p-2}u)\\ =|u|^{m-2}u+\mu|u|^{q-2}u,\;x\in\mathbb R^N,\\ u(x)>0,\;x\in\mathbb R^N,\;u(x)\in W^{1,p}(\mathbb R^N), \end{cases} \eqno{(0.1)} \] where \(a,b>0\),\({\tau},{\lambda}\geq 0,{\mu}\in \mathbb R\), \(1<p<N,p<q<m<p^\ast =\frac{pN}{N-p}\). A new existence result for (0.1) is obtained by the Nehari manifold method. This result can be regarded as an extension of the result in [Y. Li et al., J. Differ. Equations 253, No. 7, 2285–2294 (2012; Zbl 1259.35078)].

MSC:
35J40 Boundary value problems for higher-order elliptic equations
35B09 Positive solutions to PDEs
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[1] Li, Y. H.; Li, F. Y.; Shi, J. P., Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253, 2285-2294, (2012) · Zbl 1259.35078
[2] Jeanjean, L.; Le Coz, S., An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11, 813-840, (2006) · Zbl 1155.35095
[3] Kikuchi, H., Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7, 403-437, (2007) · Zbl 1133.35013
[4] Jin, J. H.; Wu, X., Infinitely many radial solutions for Kirchhoff-type problems in \(\mathbb{R}^N\), J. Math. Anal. Appl., 369, 564-574, (2010) · Zbl 1196.35221
[5] Su, J. B.; Wang, Z. Q., Sobolev type embedding and quasilinear elliptic equations with radial potentials, J. Differential Equations, 250, 223-242, (2011) · Zbl 1206.35243
[6] Su, J. B.; Wang, Z. Q.; Willem, M., Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238, 201-219, (2007) · Zbl 1220.35026
[7] Wang, L., On a quasilinear Schrödinger-Kirchhoff-type equation with radial potentials, Nonlinear Anal., 83, 58-68, (2013) · Zbl 1278.35106
[8] Badiale, M.; Serra, E., Semilinear elliptic equations for beginners, (Existence Results via the Variational Approach, (2011), Springer-Verlag London) · Zbl 1214.35025
[9] Chen, C. Y.; Kuo, Y. C.; Wu, T. F., The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250, 1876-1908, (2011) · Zbl 1214.35077
[10] Drabek, P.; Pohozaev, S. I., Positive solutions for the \(p\)-Laplacian; application of the fibering method, Proc. Roy. Soc. Edinburgh, Sect. A., 127, 703-726, (1997) · Zbl 0880.35045
[11] Pohozaev, S. I., On the method of fibering in nonlinear boundary value problems, Proc. Steklov Inst. Math., 192, 146-163, (1990) · Zbl 0734.35036
[12] Brown, K. J.; Wu, T. F., A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, 22, 1097-1114, (2009) · Zbl 1240.35569
[13] Kuzin, I.; Pohozaev, S., Entire solutions of semilinear elliptic equations, (Progress in Nonlinear Differential Equations and their Applications, vol. 33, (1997), Birkhäuser Berlin) · Zbl 0882.35003
[14] Evans, L. C., Partial differential equations, (Graduate Studies in Mathematics, vol. 19, (1998), Amer. Math. Soc.)
[15] Struwe, M., Variational methods, (2000), Springer-Verlag New York
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